UNIVERSITY
OF WROCŁAW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
44.1 43.2 43.1 42.2 42.1 41.2 41.1
40.2 40.1 39.2 39.1 38.2 38.1 37.2
37.1 36.2 36.1 35.2 35.1 34.2 34.1
33.2 33.1 32.2 32.1 31.2 31.1 30.2
30.1 29.2 29.1 28.2 28.1 27.2 27.1
26.2 26.1 25.2 25.1 24.2 24.1 23.2
23.1 22.2 22.1 21.2 21.1 20.2 20.1
19.2 19.1 18.2 18.1 17.2 17.1 16.2
16.1 15 14.2 14.1 13.2 13.1 12.2
12.1 11.2 11.1 10.2 10.1 9.2 9.1
8 7.2 7.1 6.2 6.1 5.2 5.1
4.2 4.1 3.2 3.1 2.2 2.1 1.2
1.1
 
 
WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 41, Fasc. 2,
pages 347 - 358
DOI: 10.37190/0208-4147.41.2.8
Published online 14.9.2021
 

On a conjecture about the comparability of parallel systems with respect to the convex transform order

Idir Arab
Milto Hadjikyriakou
Paulo~Eduardo Oliveira

Abstract:

We study the comparability of the lifetimes of heterogeneous parallel systems with independent exponentially distributed components. It is known that the order statistics of systems composed of two types of components may be comparable with respect to the star transform order. On what concerns the stronger convex transform order, results have been obtained only for the sample maxima assuming that one of the systems is homogeneous. We prove, under the same assumptions as for the star transform ordering, that the lifetimes of heterogeneous parallel systems are not comparable with respect to the convex transform order.

2010 AMS Mathematics Subject Classification: Primary 60E15; Secondary 60E05, 62N05.

Keywords and phrases: star transform order, convex transform order, failure rate, sign variation.

I. Arab and P. E. Oliveira, Iterated failure rate monotonicity and ordering relations within gamma and Weibull distributions, Probab. Engrg. Inform. Sci. 33 (2019), 64-80.

I. Arab and P. E. Oliveira, Iterated failure rate monotonicity and ordering relations within gamma and Weibull distributions—Corrigendum, Probab. Engrg. Inform. Sci. 32 (2018), 640-641.

I. Arab, M. Hadjikyriakou and P.E. Oliveira, Failure rate properties of parallel systems, Adv. Appl. Probab. 52 (2020), 563-587.

R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, New York, 1975.

N. Cai, W. Ni and C. Li, Some ordering properties of series and parallel systems with dependent component lifetimes, Comm. Statist. 48 (2019), 4764-4779.

J. V. Deshpande, S. C. Kochar and H. Singh, Aspects of positive ageing, J. Appl. Probab. 23 (1986), 748-758.

A. H. El-Bassiouny, On testing exponentiality against IFRA alternatives, Appl. Math. Comput. 146 (2003), 445-453.

E. Fagiuoli and F. Pellerey, New partial orderings and applications, Naval Res. Logistics 40 (1993), 829-842.

S. Kayal, Stochastic comparisons of series and parallel systems with Kumaraswamy-G distributed components, Amer. J. Math. Management Sci. 38 (2019), 1-22,

S. C. Kochar and D. D. Wiens, Partial orderings of life distributions with respect to their ageing properties, Naval Res. Logistics 34 (1987), 823-829.

S. C. Kochar and M. Xu, Comparisons of parallel systems according to the convex transform order, J. Appl. Probab. 46 (2009), 342-352.

S. C. Kochar and M. Xu, On the skewness of order statistics in multiple-outlier models, J. Appl. Probab. 48 (2011), 271-284.

C. Li and X. Li, Stochastic comparisons of parallel and series systems of dependent components equipped with starting devices, Comm. Statist. 48 (2019), 694-708.

A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Application, Academic Press, New York, 1979.

A. K. Nanda, N. K. Hazra, D. K. Al-Mutairi and M. E. Ghitany, On some generalized ageing orderings, Comm. Statist. 46 (2017), 5273-5291.

J. K. Patel, Hazard rate and other classifications of distributions, in: Encyclopedia in Statistical Sciences 3, Wiley, 1983, 590-594.

D. Sengupta, Another look at the moment bounds on reliability, J. Appl. Probab. 31 (1994), 777-787.

S. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, 2007.

H. Singh, On partial orderings, Naval Res. Logistics 36 (1989), 103-110.

T. Tossavainen, The lost cousin of the fundamental theorem of algebra, Math. Magazine 80 (2007), 290-294.

W. R. van Zwet, Convex transformations of random variables, MC Tracts 7, Amsterdam, 1964.

J. Wua, M. Wanga and X. Li, Convex transform order of the maximum of independent Weibull random variables, Statist. Probab. Lett. 156 (2020), 1-6.

Y. Zhang, X. Cai, P. Zhao and H. Wang, Stochastic comparisons of parallel and series systems with heterogeneous resilience-scaled components, Statistics 53 (2019), 126-147.

Download:    Abstract    Full text